Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

What is the best way to shunt these carriages so that each train can continue its journey?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

What happens when you try and fit the triomino pieces into these two grids?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

How many different triangles can you make on a circular pegboard that has nine pegs?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

An activity making various patterns with 2 x 1 rectangular tiles.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

How many models can you find which obey these rules?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

How many different rhythms can you make by putting two drums on the wheel?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

These practical challenges are all about making a 'tray' and covering it with paper.

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Can you find all the different ways of lining up these Cuisenaire rods?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Investigate the different ways you could split up these rooms so that you have double the number.

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

How many trains can you make which are the same length as Matt's, using rods that are identical?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.